Vibration analysis of a double layer microshell utilizing a modified couple stress theory

Document Type : Research Paper


Faculty of Engineering, Department of Mechanics, Imam Khomeini International University


In this paper, dynamic modeling of a double layer cylindrical functionally graded (FG) microshell is considered. Modeling is based on the first-order shear deformation theory (FSDT), and the equations of motion are derived using the Hamilton's principle. It assumes that functionally graded length scale parameter changes along the thickness. Generalized differential quadrature method (GDQM) is used to discretize the model and solve the problem.  In this research the size effect is investigated using a new modified couple stress theory (MCST) which has been considered for the first time in the present study. The accuracy of the presented model is validated with some cases in the literature. Considering the microshell as double layer and utilizing the MCST in addition to considering the various boundary conditions are the novelty of this study. Furthermore, the effects of length, thickness, FG power index, Winkler and Pasternak coefficients and shear correction factor on the natural frequency of double layer cylindrical FG microshell are studied.


Main Subjects

 [1] Haddadpour, H., Mahmoudkhani, S., and Navazi, H., "Free Vibration Analysis of Functionally Graded Cylindrical Shells Including Thermal Effects", Thin-Walled Structures,Vol. 45, pp. 591-599, (2007)
[2] Farid, M., Zahedinejad, P., and Malekzadeh, P., "Three-dimensional Temperature Dependent Free Vibration Analysis of Functionally Graded Material Curved Panels Resting on Two-parameter Elastic Foundation using a Hybrid Semi-analytic, Differential Quadrature Method", Materials & Design,Vol. 31, pp. 2-13, (2010).
[3] Rahaeifard, M., Kahrobaiyan, M., and Ahmadian, M., "Sensitivity Analysis of Atomic Force Microscope Cantilever made of Functionally Graded Materials", in ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 539-544, San Diego, California, USA, (2009).
[4] Lee, Z., Ophus, C., Fischer, L.M., Nelson-Fitzpatrick, N., Westra, K.L., Evoy, S., Radmilovic, V., Dahmen, U., and Mitlin, D.,"Metallic NEMS Components Fabricated from Nanocomposite Al–Mo Films", Nanotechnology, Vol. 17, pp. 3063, (2006).
[5] Ghasemabadian, M.A., and Kadkhodayan, M., "Investigation of Buckling Behavior of Functionally Graded Piezoelectric (FGP) Rectangular Plates under Open and Closed Circuit Conditions", Structural Engineering and Mechanics,Vol. 60, No. 2, pp. 271-299, (2016).
[6] Koiter, W., "Couple Stresses in the Theory of Elasticity," Proc. Koninklijke Nederl. Akaad. Van Wetensch,Vol. 67, (1964).
[7] Mindlin, R. D., "Micro-structure in Linear Elasticity", Archive for Rational Mechanics and Analysis, Vol. 16, pp. 51-78, (1964).
 [8] Asghari, M., Kahrobaiyan, M., Rahaeifard, M., and Ahmadian, M., "Investigation of the Size Effects in Timoshenko Beams Based on the Couple Stress Theory", Archive of Applied Mechanics,Vol. 81, pp. 863-874, (2011).
[9] Yang, F., Chong, A.C.M., Lam, D.C.C., and Penger, T., "Couple Stress Based Strain Gradient Theory for Elasticity", International Journal of Solids and Structures, Vol. 39, pp. 2731-2743, (2002).
[11] Miandoab, E. M., Pishkenari, H. N.,  Yousefi-Koma, A., and Hoorzad, H., "Polysilicon Nano-beam Model Based on Modified Couple Stress and Eringen’s Nonlocal Elasticity Theories", Physica E: Low-dimensional Systems and Nanostructures,Vol. 63, pp. 223-228, (2014).
[12] Karami, H.,  and Farid, M., "A New Formulation to Study In-plane Vibration of Curved Carbon Nanotubes Conveying Viscous Fluid", Journal of Vibration and Control,Vol. 21, pp. 2360-2371, (2015).
[13] Choi, J., Song, O., and Kim, S. k., "Nonlinear Stability Characteristics of Carbon Nanotubes Conveying Fuids", Acta Mechanica,Vol. 224, pp. 1383-1396, (2013).
[14] Chang, T. P., "Axial Vibration of Non-uniform and Non-homogeneous Nanorods Based on Nonlocal Elasticity Theory", Applied Mathematics and Computation,Vol. 219, pp. 4933-4941, (2013).

[15] Zhang, H., Deng, Q.T., and Li, S.H., "Vibration of a Single-walled Carbon Nanotube Embedded in an Elastic Medium under a Moving Internal Nanoparticle", Applied Mathematical Modelling, Vol. 37, Issues. 10–11, pp. 6940-6951, (2013).


[16] Ghadiri, M.,  and Shafiei, N., "Nonlinear Bending Vibration of a Rotating Nanobeam Based on Nonlocal Eringen’s Theory using Differential Quadrature Method", Microsystem Technologies, pp. 1-15, (2015).
[17] Habibi, M., Taghdir, A., and Safarpour, H., "Stability Analysis of an Electrically Cylindrical Nanoshell Reinforced with Graphene Nanoplatelets", Composites Part B: Engineering, Vol. 175, pp. 107-125, (2019).
 [18] Ghadiri, M., and Safarpour, H., "Free Vibration Analysis of Embedded Magneto-electro-thermo-elastic Cylindrical Nanoshell Based on the Modified Couple Stress Theory", Applied Physics A,Vol. 122, pp. 833, (2016).